Stable and unstable periodic orbits and their bifurcations in the nonlinear dynamical system with a fixed point vortex in a periodic flow

Abstract

In this paper, periodic orbits in the nonlinear dynamical system with a fixed point vortex in a periodic flow are investigated. Under the influence of periodic perturbations in the phase space, an infinite number of nonlinear resonances with elliptic and hyperbolic periodic orbits arise. It is shown that these orbits exist even with completely destroyed resonant islands. In the perturbed system, all periodic orbits with periods up to T=4T0, where T0 is period of perturbation, are found. The existence of nonlinear resonances of the KAM and non-KAM nature is shown, and a genetic relationship between different orbits is established. All elliptic orbits are destroyed with an increase of perturbation by the universal cascade of period doubling. It is shown that the rates of cascades for different orbits have close values and are consistent with the value of the Feigenbaum constant for two-dimensional conservative mappings. The complex interaction of hyperbolic orbits of the secondary resonances with the elliptic orbit of the primary resonance is demonstrated. It is shown that in addition to the universal cascade of period doubling, other bifurcation scenarios common to different orbits are possible.